System and method for rendering images using a Russian Roulette methodology for evaluating global illumination

ABSTRACT

A computer graphics system generate a pixel value for a pixel in an image to simulate global illumination represented by an evaluation of an unknown function f of the form f(x)=g(x)+∫ 0   1 K(x,y)f(y)dy,g(x) and K(x,y) known functions, with K(x,y) a “kernel” including a function associated with at least two colors. An estimator generator module generates “N” estimators f (n)   lds,RR (x) as  
                   f   _       lds   ,   RR       (   n   )            (   x   )       =                  g        (   x   )       +       K        (     x   ,     ξ   1     (   n   )         )            S   1     (   n   )         +                                  K        (     x   ,     ξ   2     (   n   )         )            S   2     (   n   )         +                                    K        (     x   ,     ξ   3     (   n   )         )            S   3     (   n   )         +   ⋯                ,                 or                   f     lds   ,   RR       (   n   )            (   x   )       =                  g        (   x   )       +         T   1     (   n   )            (   x   )            g        (     ξ   1     (   n   )       )         +                                    T   2     (   n   )            (   x   )            g        (     ξ   2     (   n   )       )         +                                      T   3     (   n   )            (   x   )            g        (     ξ   3     (   n   )       )         +   ⋯                ,                 where             S   1     (   n   )       :=     g        (     ξ   1     (   n   )       )         ,     
            S     j   +   1       (   n   )       :=         Θ        (       M       K        (       ξ     j   +   1     n     ,     ξ   j     (   n   )         )       ,     S   j     (   n   )           -     ξ   j     ′        (   n   )           )         M       K        (       ξ     j   +   1     n     ,     ξ   j   n       )       ,     S   j     (   n   )                  K        (       ξ     j   +   1     n     ,     ξ   j   n       )            S   j     (   n   )           ,     
        and                 T   1     (   n   )            (   x   )       :=         Θ        (       M       K        (     x   ,     ξ   1     (   n   )         )       ,   1       -     ξ   1     ′        (   n   )           )         M       K        (     x   ,     ξ   1     (   n   )         )       ,   1              K        (     x   ,     ξ   1     (   n   )         )           ,     
              T     j   +   1       (   n   )            (   x   )       :=         Θ        (       M       K        (       ξ   j     (   n   )       ,     ξ     j   +   1       (   n   )         )       ,       T   j     (   n   )            (   x   )           -     ξ     j   +   1       ′        (   n   )           )         M       K        (       ξ   j     (   n   )       ,     ξ     j   +   1       (   n   )         )       ,       T   j     (   n   )            (   x   )                  K        (       ξ   j     (   n   )       ,     ξ     j   +   1       (   n   )         )                 with           M     A   ,   B       :=     {                   ∫   0   ∞            A   λ          B   λ             λ             ∫   0   ∞            B   λ             λ           ,             if                     ∫   0   ∞            B   λ             λ           &gt;   0               0   ,             if                     ∫   0   ∞            B   λ             λ           =   0           ,                     
 
     Θ(x) representing the Heaviside function, ξ 1   (n) , ξ′ 1   (n) , . . . , ξ L   (n) , ξ′ L   (n)  components of a multi-dimensional strictly deterministic low-discrepancy sequence and “n” a sequence index. A luminance value module generates a luminance value for the point in the scene as the average of the estimator values generated by the estimator generator module. A pixel value generator module uses the luminance value generated for the point in the scene in generating the pixel value.

INCORPORATION BY REFERENCE

[0001] U.S. patent application Ser. No. 08/880,418, filed Jun. 23, 1997,in the names of Martin Grabenstein, et al., entitled “System And MethodFor Generating Pixel Values For Pixels In An Image Using StrictlyDeterministic Methodologies For Generating Sample Points,” (hereinafterreferred to as the Grabenstein application) assigned to the assignee ofthis application, incorporated by reference.

FIELD OF THE INVENTION

[0002] The invention relates generally to the field of computergraphics, and more particularly to systems and methods for generatingpixel values for pixels in an image being rendered usingstrictly-deterministic low-discrepancy sequences to provide samplepoints for generating estimates of values of integrals representing thepixel values.

BACKGROUND OF THE INVENTION

[0003] In computer graphics, a computer is used to generate digital datathat represents the projection of surfaces of objects in, for example, athree-dimensional scene, illuminated by one or more light sources, ontoa two-dimensional image plane, to simulate the recording of the sceneby, for example, a camera. The camera may include a lens for projectingthe image of the scene onto the image plane, or it may comprise apinhole camera in which case no lens is used. The two-dimensional imageis in the form of an array of picture elements (which are variabletermed “pixels” or “Pels”), and the digital data generated for eachpixel represents the color and luminance of the scene as projected ontothe image plane at the point of the respective pixel in the image plane.The surfaces of the objects may have any of a number of characteristics,including shape, color, specularity, texture, and so forth, which arepreferably rendered in the image as closely as possible, to provide arealistic-looking image.

[0004] Generally, the contributions of the light reflected from thevarious points in the scene to the pixel value representing the colorand intensity of a particular pixel are expressed in the form of the oneor more integrals of relatively complicated functions. One renderingmethodology, “global illumination,” includes a class of optical effects,such as indirect illumination, reflections off surfaces with varioustypes of characteristics, such as diffuse, glossy and specular surfaces,caustics and color bleeding, which the are simulated in generating animage. In global illumination, a “rendering equation” is solved, whichhas a general form

f(x)=g(x)+∫₀ ¹ K(x,y)f(y)dy  (1)

[0005] where “g” and “K” are known functions (“K” is sometimes referredto as the “kernel” of the integral equation), and “f” is an unknownfunction. Generally, “f(x)” represents the luminance at a particularpoint in the image from a particular direction and “g(x)” representsambient illumination. The function “K,” is typically a complex functiontypically including factors such as characteristics of the objects'surfaces, such as the degree to which they are glossy, diffuse and/orspecular, the angular relationships of surfaces with respect to eachother, whether one surface is visible from another, and so forth. Inconnection with one aspect of simulation using ray tracing, in which animage as viewed by an observer is simulated by tracing rays,representing photons, between a light source and the eye of theobserver, the unknown function “f” can be obtained as $\begin{matrix}\begin{matrix}{{f(x)} = \quad {{g(x)} + {\int_{0}^{1}{{K\left( {x,x_{1}} \right)}{g\left( x_{1} \right)}{x_{1}}}} +}} \\{\quad {{\int_{0}^{1}{\int_{0}^{1}{{K\left( {x,x_{1}} \right)}{K\left( {x_{1},x_{2}} \right)}{g\left( x_{2} \right)}{x_{1}}{x_{2}}}}} +}} \\{\quad {{\int_{0}^{1}{\int_{0}^{1}{\int_{0}^{1}{{K\left( {x,x_{1}} \right)}{K\left( {x_{1},x_{2}} \right)}{K\left( {x_{2},x_{3}} \right)}{g\left( x_{3} \right)}{x_{1}}{x_{2}}{x_{3}}}}}},}}\end{matrix} & (2)\end{matrix}$

[0006] if the series converges. The series is guaranteed to converge iflim sup_(n→∞) ^(n){square root}{square root over (∥K^(n)∥)}<1.

[0007] As described in the Grabenstein application, an approximation ofthe unknown function “f” is obtained as follows. Defining the Heavisidefunction Θ(x) as $\begin{matrix}{{\Theta (x)} = \left\{ {\begin{matrix}{1,} & {{{for}\quad x} > 0} \\{0,} & {{{for}\quad x} \leq 0}\end{matrix},} \right.} & (3)\end{matrix}$

[0008] which will be referred to as a “Russian Roulette operator,” the“n-th” estimate of equation (2) is given by $\begin{matrix}\begin{matrix}{f_{{lds},{RR}}^{(n)} = \quad {{g(x)} + {{\Theta \left( {{K\left( {x,\xi_{1}^{(n)}} \right)} - \xi_{1}^{\prime {(n)}}} \right)}{g\left( \xi_{1}^{(n)} \right)}} +}} \\{\quad {{{\Theta \left( {{K\left( {x,\xi_{1}^{(n)}} \right)} - \xi_{1}^{\prime {(n)}}} \right)}{\Theta \left( {{K\left( {\xi_{1}^{(n)},\xi_{2}^{(n)}} \right)} - \xi_{2}^{\prime {(n)}}} \right)}{g\left( \xi_{2}^{(n)} \right)}} +}} \\{\quad {{{{\Theta \left( {{K\left( {x,\xi_{1}^{(n)}} \right)} - \xi_{1}^{\prime {(n)}}} \right)}{\Theta \left( {{K\left( {\xi_{1}^{(n)},\xi_{2}^{(n)}} \right)} - \xi_{2}^{\prime {(n)}}} \right)}{\Theta \left( {{K\left( {\xi_{2}^{(n)},\xi_{3}^{(n)}} \right)} - \xi_{3}^{\prime {(n)}}} \right)}{g\left( \xi_{3}^{(n)} \right)}} + \cdots}\quad,}}\end{matrix} & (4)\end{matrix}$

[0009] where ξ₁ ^((n)), . . . , ξ_(L) ^((n)) and ξ′₁ ^((n)), . . . ,ξ′_(L) ^((n)) comprise “s”-dimensional low-discrepancy sequences, suchas Halton sequences, which provide sample points in the s-dimensionalunit cube [1,0)^(s) for the respective estimation. Halton sequences aredescribed in the Grabenstein application. The series in equation (4),which corresponds to equation (27) in the Grabenstein application, isused in connection with ray tracing in the direction from the imagetoward the light source(s), which is sometimes referred to aseye-to-light source ray tracing, where the image to be renderedsimulates the image of a scene as would be seen by the eye of a personviewing the scene. In equation (4), the first summand of the series onthe right-hand side (that is, the “g(x)” term) represents ambientillumination, the next summand (that is, the first “Θ” term) representsdirect illumination, the next summand represents depth one reflection orrefraction in the direction from the eye toward the light source, thenext summand represents depth two reflection or refraction in thedirection from the eye toward the light source, and so forth. TheRussian Roulette operator essentially operates as a Russian Roulettecheck, and will guarantee that the series in equation (4) willeventually terminate at some summand, since eventually the value of oneof the Russian Roulette operators Θ will equal zero, and that RussianRoulette operator Θ would also be in all subsequent terms of the series.Note that, in equation (4)

[0010] (i) each Russian Roulette check requires the evaluation of thekernel “K” at “x” and, respectively, ξ₁ ^((n)), . . . , ξ_(L) ^((n)),and

[0011] (ii) for the respective summands in equation (18), the value offunction “g” that multiplies Θ in the summands is evaluated at differentsample points ξ₁ ^((n)), . . . , ξ_(L) ^((n)).

[0012] As an alternative, if instead of using the image-to-light sourceray tracing, the ray tracing is performed from the light source towardthe image (which is sometimes referred to as light-to-eye ray tracing),the “n-th” estimation can be performed such that the value of thefunction “g” that multiply the Russian Roulette operator Θ in thesummands is evaluated at one sample point, namely ξ₁ ^((n)). In lightsource-to-eye ray tracing, instead of equation (2), the unknown function“f” can be obtained as $\begin{matrix}\begin{matrix}{{f(x)} = \quad {{g(x)} + {\int_{0}^{1}{{K\left( {x,x_{1}} \right)}{g\left( x_{1} \right)}{x_{1}}}} +}} \\{\quad {{\int_{0}^{1}{\int_{0}^{1}{{K\left( {x,x_{2}} \right)}{K\left( {x_{2},x_{1}} \right)}{g\left( x_{1} \right)}{x_{1}}{x_{2}}}}} +}} \\{\quad {{\int_{0}^{1}{\int_{0}^{1}{\int_{0}^{1}{{K\left( {x,x_{3}} \right)}{K\left( {x_{3},x_{2}} \right)}{K\left( {x_{2},x_{1}} \right)}{g\left( x_{1} \right)}{x_{1}}{x_{2}}{x_{3}}}}}},}}\end{matrix} & (5)\end{matrix}$

[0013] which, in turn, can be estimated using $\begin{matrix}\begin{matrix}{{{\overset{\_}{f}}_{{lds},{RR}}^{(n)}(x)} = \quad {{g(x)} + {{K\left( {x,\xi_{1}^{(n)}} \right)}{g\left( \xi_{1}^{(n)} \right)}} +}} \\{\quad {{{K\left( {x,\xi_{2}^{(n)}} \right)}{\Theta \left( {{K\left( {\xi_{2}^{(n)},\xi_{1}^{(n)}} \right)} - \xi_{1}^{\prime {(n)}}} \right)}{g\left( \xi_{1}^{(n)} \right)}} +}} \\{\quad {{K\left( {x,\xi_{3}^{(n)}} \right)}{\Theta\left( {{K\left( {\xi_{3}^{(n)},\xi_{2}^{(n)}} \right)} -} \right.}}} \\{{\left. \quad \xi_{2}^{\prime {(n)}} \right){\Theta \left( {{K\left( {\xi_{2}^{(n)},\xi_{1}^{(n)}} \right)} - \xi_{1}^{\prime {(n)}}} \right)}{g\left( \xi_{1}^{(n)} \right)}} + {\cdots \quad.}}\end{matrix} & (6)\end{matrix}$

[0014] As with equation (4), the first summand of the series on theright-hand side (that is, the “g(x)” term) represents ambientillumination, and the next summand (that is, the first “Θ” term)represents direct illumination. In contrast with equation (4), the nextsummand represents depth one reflection or refraction in the directionfrom the light source toward the eye, the next summand represents depthtwo reflection or refraction in the direction from the light sourcetoward the eye, and so forth. As with equation (4), the series inequation (6) will truncate since eventually the value of one of theRussian Roulette operators Θ will equal zero, and that Russian Rouletteoperator Θ would also be in all subsequent terms of the series. Acomparison of equation (6) with equation (4) reveals that

[0015] (i) since the top line in equation (6) does not involve a RussianRoulette check, one fewer Russian Roulette check will typically berequired than in equation (4); and

[0016] (ii) function “g” need only be evaluated at one sample point,namely, ξ₁ ^((n)), reducing the amount of processing time required togenerate each summand.

[0017] The methodology described above in connection with equations (1)through (6) generally assumes that the color is monochromatic. Theinvention provides an improved Russian Roulette methodology that allowsfor use in connection with a variety of colors and characteristics ofsurfaces of objects in the scene.

SUMMARY OF THE INVENTION

[0018] The invention provides a new and improved system and method forgenerating pixel values for pixels in an image being rendered usingstrictly-deterministic low-discrepancy sequences to provide samplepoints for generating estimates of values of integrals representing thepixel values.

[0019] In brief summary, the invention in one aspect provides a computergraphics system for generating a pixel value for a pixel in an image,the pixel being representative of a point {right arrow over (x)} in ascene. The computer graphics system generates the pixel value tosimulate global illumination for a plurality of colors, globalillumination being represented by an evaluation of a function f(x), thefunction f(x) having the form f(x)=g(x)+∫₀ ¹K(x,y)f(y)dy, where f(x)(and similarly f(y)) is an unknown function, and g(x) and K(x,y) areknown functions, with K(x,y) serving as a “kernel” of the integral, thekernel K(x,y) including a function associated with at least two colors.The computer graphics system comprises an estimator generator module, aluminance value module and a pixel value generator. The estimatorgenerator module is configured to generate a selected number “N” ofestimators f^((n)) _(lds,RR)(x) as $\begin{matrix}{{{\overset{\_}{f}}_{{lds},{RR}}^{(n)}(x)} = \quad {{g(x)} + {{K\left( {x,\xi_{1}^{(n)}} \right)}S_{1}^{(n)}} +}} \\{\quad {{{K\left( {x,\xi_{2}^{(n)}} \right)}S_{2}^{(n)}} +}} \\{\quad {{{{K\left( {x,\xi_{3}^{(n)}} \right)}S_{3}^{(n)}} + \cdots}\quad,}}\end{matrix}$

[0020] where S₁ ^((n)) are defined recursively as${S_{1}^{(n)}:={g\left( \xi_{1}^{(n)} \right)}},{S_{j + 1}^{(n)}:={\frac{\Theta \left( {M_{{K{({\xi_{j + 1}^{n},\xi_{j}^{(n)}})}},S_{j}^{(n)}} - \xi_{j}^{\prime {(n)}}} \right)}{M_{{K{({\xi_{j + 1}^{n},\xi_{j}^{n}})}},S_{j}^{(n)}}}{K\left( {\xi_{j + 1}^{n},\xi_{j}^{n}} \right)}S_{j}^{(n)}}},{with}$$M_{A,B}:=\left\{ {\begin{matrix}{\frac{\int_{0}^{\infty}{A_{\lambda}B_{\lambda}{\lambda}}}{\int_{0}^{\infty}{B_{\lambda}{\lambda}}},} & {{{if}\quad {\int_{0}^{\infty}{B_{\lambda}{\lambda}}}} > 0} \\{0,} & {{{if}\quad {\int_{0}^{\infty}{B_{\lambda}{\lambda}}}} = 0}\end{matrix},} \right.$

[0021] and where Θ(x) represents the Heaviside function, and ξ₁ ^((n)),ξ′₁ ^((n)), ξ₂ ^((n)), ξ′₂ ^((n)), . . . , ξ_(L) ^((n)), ξ′_(L) ^((n)),represent components of a predetermined multi-dimensional strictlydeterministic low-discrepancy sequence and where “n” denotes a sequenceindex, the estimator generator generating successive terms for eachestimator f^((n)) _(lds,RR)(x) until it generates a term having thevalue zero. The luminance value module is configured to generate aluminance value for the point in the scene as the average of saidestimator values generated by said estimator generator module. The pixelvalue generator module is configured to use the luminance valuegenerated for the point in the scene in generating the pixel value.

[0022] In another aspect, the invention provides a computer graphicssystem for generating a pixel value for a pixel in an image, the pixelbeing representative of a point {right arrow over (x)} in a scene. Thecomputer graphics system generates the pixel value to simulate globalillumination for a plurality of colors, global illumination beingrepresented by an evaluation of a function f(x) over a sphere centeredat the point in the scene, the function f(x) having the formf(x)=g(x)+∫₀ ¹K(x,y)f(y)dy, where f(x) (and similarly f(y)) is anunknown function, and g(x) and K(x,y) are known functions, with K(x,y)serving as a “kernel” of the integral, the kernel K(x,y) including afunction associated with at least two colors. The computer graphicssystem comprises an estimator generator module, a luminance value moduleand a pixel value generator. The estimator generator module isconfigured to generate a selected number “N” of estimators f^((n))_(lds,RR)(x) as $\begin{matrix}{{f_{{lds},{RR}}^{(n)}(x)} = \quad {{g(x)} + {{T_{1}^{(n)}(x)}{g\left( \xi_{1}^{(n)} \right)}} +}} \\{\quad {{{T_{2}^{(n)}(x)}{g\left( \xi_{2}^{(n)} \right)}} +}} \\{\quad {{{{T_{3}^{(n)}(x)}{g\left( \xi_{3}^{(n)} \right)}} + \cdots}\quad,}}\end{matrix}$where  S_(i)^((n))  are  defined  recursively  as${{T_{1}^{(n)}(x)}:={\frac{\Theta \left( {M_{{K{({x,\xi_{1}^{(n)}})}},1} - \xi_{1}^{\prime {(n)}}} \right)}{M_{{K{({x,\xi_{1}^{(n)}})}},1}}{K\left( {x,\xi_{1}^{(n)}} \right)}}},{{T_{j + 1}^{(n)}(x)}:={\frac{\Theta \left( {M_{{K{({\xi_{j}^{(n)},\xi_{j + 1}^{(n)}})}},{T_{j}^{(n)}{(x)}}} - \xi_{j + 1}^{\prime {(n)}}} \right)}{M_{{K{({\xi_{j}^{(n)},\xi_{j + 1}^{(n)}})}},{T_{j}^{(n)}{(x)}}}}{K\left( {\xi_{j}^{(n)},\xi_{j + 1}^{(n)}} \right)}}}$with $M_{A,B}:=\left\{ {\begin{matrix}{\frac{\int_{0}^{\infty}{A_{\lambda}B_{\lambda}{\lambda}}}{\int_{0}^{\infty}{B_{\lambda}{\lambda}}},} & {{{if}\quad {\int_{0}^{\infty}{B_{\lambda}{\lambda}}}} > 0} \\{0,} & {{{if}\quad {\int_{0}^{\infty}{B_{\lambda}{\lambda}}}} = 0}\end{matrix},} \right.$

[0023] and where Θ(x) represents the Heaviside function, and ξ₁ ^((n)),ξ′₁ ^((n)), ξ₂ ^((n)), ξ′₂ ^((n)), . . . , ξ_(L) ^((n)), ξ′_(L) ^((n)),represent components of a predetermined multi-dimensional strictlydeterministic low-discrepancy sequence and where “n” denotes a sequenceindex, the estimator generator generating successive terms for eachestimator f^((n)) _(lds,RR)(x) until it generates a term having thevalue zero. The luminance value generator is configured to generate aluminance value for the point in the scene as the average of saidestimator values generated by said estimator generator module. The pixelvalue generator module is configured to use the luminance valuegenerated for the point in the scene in generating the pixel value.

BRIEF DESCRIPTION OF THE DRAWINGS

[0024] This invention is pointed out with particularity in the appendedclaims. The above and further advantages of this invention may be betterunderstood by referring to the following description taken inconjunction with the accompanying drawings, in which:

[0025]FIG. 1 depicts an illustrative computer graphics systemconstructed in accordance with the invention.

DETAILED DESCRIPTION OF AN ILLUSTRATIVE EMBODIMENT

[0026] The invention provides a computer graphic system and method forgenerating pixel values for pixels in an image of a scene, which makesuse of a global illumination methodology that allows for use inconnection with a variety of colors and characteristics of surfaces ofobjects in the scene. FIG. 1 attached hereto depicts an illustrativecomputer system 10 that makes use of such a global illuminationmethodology. With reference to FIG. 1, the computer system 10 in oneembodiment includes a processor module 11 and operator interfaceelements comprising operator input components such,as a keyboard 12Aand/or a mouse 12B (generally identified as operator input element(s)12) and an operator output element such as a video display device 13.The illustrative computer system 10 is of the conventionalstored-program computer architecture. The processor module 11 includes,for example, one or more processor, memory and mass storage devices,such as disk and/or tape storage elements (not separately shown), whichperform processing and storage operations in connection with digitaldata provided thereto. If the processor module 11 includes a pluralityof processor devices, the respective processor devices may be configuredto process various portions of a single task in parallel, in which casethe task may be executed more quickly than otherwise. The operator inputelement(s) 12 are provided to permit an operator to input informationfor processing. The video display device 13 is provided to displayoutput information generated by the processor module 11 on a screen 14to the operator, including data that the operator may input forprocessing, information that the operator may input to controlprocessing, as well as information generated during processing. Theprocessor module 11 generates information for display by the videodisplay device 13 using a so-called “graphical user interface” (“GUI”),in which information for various applications programs is displayedusing various “windows.” Although the computer system 10 is shown ascomprising particular components, such as the keyboard 12A and mouse 12Bfor receiving input information from an operator, and a video displaydevice 13 for displaying output information to the operator, it will beappreciated that the computer system 10 may include a variety ofcomponents in addition to or instead of those depicted in FIG. 1.

[0027] In addition, the processor module 11 includes one or more networkports, generally identified by reference numeral 14, which are connectedto communication links which connect the computer system 10 in acomputer network. The network ports enable the computer system 10 totransmit information to, and receive information from, other computersystems and other devices in the network. In a typical network organizedaccording to, for example, the client-server paradigm, certain computersystems in the network are designated as servers, which store data andprograms (generally, “information”) for processing by the other, clientcomputer systems, thereby to enable the client computer systems toconveniently share the information. A client computer system which needsaccess to information maintained by a particular server will enable theserver to download the information to it over the network. Afterprocessing the data, the client computer system may also return theprocessed data to the server for storage. In addition to computersystems (including the above-described servers and clients), a networkmay also include, for example, printers and facsimile devices, digitalaudio or video storage and distribution devices, and the like, which maybe shared among the various computer systems connected in the network.The communication links interconnecting the computer systems in thenetwork may, as is conventional, comprise any convenientinformation-carrying medium, including wires, optical fibers or othermedia for carrying signals among the computer systems. Computer systemstransfer information over the network by means of messages transferredover the communication links, with each message including informationand an identifier identifying the device to receive the message.

[0028] The invention provides an improved Russian Roulette methodologyto provide for multichromatic color and various surface characteristics,such as glossy, diffuse and specular, of objects in a scene, in an imagerendered using the global illumination methodology. As noted above,global illumination involves solving a “rendering equation,” which has ageneral form

f(x)=g(x)+∫₀ ¹ K(x,y)f(y)dy  (13),

[0029] were “g” and kernel “K” are known functions and “f” is an unknownfunction. Generally the luminance of a particular color can berepresented as an integral over a spectrum with a weight I_(λ):∫₀^(∞)I_(λ)y_(λ)dλ, where “λ” refers to wavelength and “y_(λ)” refers tothe spectral radiance, that is, the radiant flux, at wavelength λ, perunit wavelength interval and “I_(λ)” is the photopic spectral luminousefficiency, which is a weight with which the human eye perceives lighthaving the respective wavelength λ. For convenience, it will be assumedthat ∫₀ ^(∞)I_(λ)dλ=1, that is, that the luminance over the entirespectrum is normalized. In practice, typically the luminance intensity“I,” instead of being a continuous function of wavelength λ, isrepresented by a predetermined set of color channels. Typically threechannels, such as separate red, blue, and green channels, are used. Inthat case, the luminance intensity “I” can be written in the formI=c_(r)δ_(r)+c_(g)δ_(g)+c_(b)δ_(b), where “c_(r),” “c_(g),” and “c_(b)”are coefficients that are used for the respective color channels by aparticular standard (subscripts “r,” “g” and “b” may represent “red,”“green” and “blue,” respectively, or any other set of colors that isselected for use by the standard), and “δ_(r),” δ_(g)” and “δ_(b)” aredelta functions having the value “one” if a color associated with therespective color channel is present, and otherwise “zero.”

[0030] An intensity change factor M_(A,B) for a functional B by applyinganother functional A is defined $\begin{matrix}{M_{A,B}:=\left\{ {\begin{matrix}{\frac{\int_{0}^{\infty}{A_{\lambda}B_{\lambda}{\lambda}}}{\int_{0}^{\infty}{B_{\lambda}{\lambda}}},} & {{{if}\quad {\int_{0}^{\infty}{B_{\lambda}{\lambda}}}} > 0} \\{0,} & {{{if}\quad {\int_{0}^{\infty}{B_{\lambda}{\lambda}}}} = 0}\end{matrix},} \right.} & (14)\end{matrix}$

[0031] where “A” in equation (8) will constitute the kernel “K”(reference equation (7)) at points (x,y). Under the assumption that thevalue of the norm of K is less than or equal to “one,” which, incomputer graphics, will represent the fact that the energy of a photondoes not increase after the photon is emitted by the light source(s),M_(K(x,y),B)≦1 for all (x,y). If “B” is also a function of “y”, that is,B=t(y), as is the case with “f(y)” in equation (7), then $\begin{matrix}{{{K\left( {x,y} \right)}{t(y)}} = {\int_{0}^{1}{\left( {\frac{{K\left( {x,y} \right)}{t(y)}}{M_{{K{({x,y})}},{t{(y)}}}}{\Theta \left( {M_{{K{({x,y})}},{t{(y)}}} - } \right)}} \right){}}}} & (15)\end{matrix}$

[0032] Defining recursively $\begin{matrix}{{S_{1}^{(n)}:={g\left( \xi_{1}^{(n)} \right)}},{S_{j + 1}^{(n)}:={\frac{\Theta \left( {M_{{K{({\xi_{j + 1}^{n},\xi_{j}^{n}})}},S_{j}^{(n)}} - \xi_{j}^{\prime {(n)}}} \right)}{M_{{K{({\xi_{j + 1}^{n},\xi_{j}^{n}})}},S_{j}^{(n)}}}{K\left( {\xi_{j + 1}^{n},\xi_{j}^{n}} \right)}S_{j}^{(n)}}}} & (16)\end{matrix}$

[0033] and using equation (9), equation (6), the equation for estimatingthe evaluation of equation (15) for light source-to-eye ray tracing, canbe generalized to accommodate a color spectrum as $\begin{matrix}\begin{matrix}{{{\overset{\_}{f}}_{{lds},{RR}}^{(n)}(x)} = \quad {{g(x)} + {{K\left( {x,\xi_{1}^{(n)}} \right)}S_{1}^{(n)}} +}} \\{\quad {{{K\left( {x,\xi_{2}^{(n)}} \right)}S_{2}^{(n)}} +}} \\{\quad {{{K\left( {x,\xi_{3}^{(n)}} \right)}S_{3}^{(n)}} + {\cdots \quad.}}}\end{matrix} & (17)\end{matrix}$

[0034] Equation (11) is a generalization of equation (6), with thekernel “K” adjusting the color spectrum, while keeping the luminanceintensity constant. For monochromatic images, the value ofM_(K(ξ_(j + 1)^((n)), ξ_(j)^((n))), S_(j)^((n)))

[0035] in equation (21) corresponds to K(ξ_(j + 1)^((n)), ξ_(j)^((n))),

[0036] in which caseS_(j + 1)^((n)) = Θ(K(ξ_(j + 1)^((n)), ξ_(j)^((n))) − ξ_(j)^(′(n)))g(ξ₁^((n)))

[0037] and equation (11) reduces to equation (6). The S_(J) ^((n))multipliers in the summands in equation (11) essentially representdescriptions of the flux of photons that are to be incident on a surfaceof an object in a scene, prior to their interaction with a respectivesurface of an object in the scene onto which they are incident. TheK(x,ξ_(j) ^((n))) multiplicands, in turn, represent the interactions ofthe photon flux with the respective surface, in particular how therespective surface reflects, absorbs, blocks, and so forth, respectiveones of the photons of the various wavelengths that comprise the fluxthat is incident on the respective surface.

[0038] As noted above, equations (8) through (11) assume that the normof kernel “K” is less than or equal to one. It will be appreciated that,if the value of the norm of the kernel “K” is not less than or equal to“one,” that is, if M_(K(ξ_(j + 1)^((n)), ξ_(j)^((n))), S_(j)^((n))) > 1

[0039] for at least one set of values of “j” and “n,”M_(K(ξ_(j + 1)^((n)), ξ_(j)^((n))), S_(j)^((n)))

[0040] can be split into two branches⌊M_(K(ξ_(j + 1)^((n)), ξ_(j)^((n))), S_(j)^((n)))⌋  or  ⌊M_(K(ξ_(j + 1)^((n)), ξ_(j)^((n))), S_(j)^((n))) + 1⌋

[0041] using $\begin{matrix}{{\Theta \left( {M_{{K{({\xi_{j + 1}^{(n)},\xi_{j}^{(n)}})}},S_{j}^{(n)}} - \left\lfloor M_{{K{({\xi_{j + 1}^{(n)},\xi_{j}^{(n)}})}},S_{j}^{(n)}} \right\rfloor - ^{(n)}} \right)},} & (18)\end{matrix}$

[0042] for some splitting control value ζ^((n)), to control thebranching decision.

[0043] Equation (4), which is used in estimating the evaluation ofequation (1) for eye-to-light source ray tracing, can be generalized toaccommodate a color spectrum in a similar way. Defining recursively$\begin{matrix}{{{T_{1}^{(n)}(x)}:={\frac{\Theta \left( {M_{{K{({x,\xi_{1}^{(n)}})}},1} - \xi_{1}^{\prime {(n)}}} \right)}{M_{{K{({x,\xi_{1}^{(n)}})}},1}}{K\left( {x,\xi_{1}^{(n)}} \right)}}}{{T_{j + 1}^{(n)}(x)}:={\frac{\Theta \left( {M_{{K{({\xi_{j}^{(n)},\xi_{j + 1}^{(n)}})}},{T_{j}^{(n)}{(x)}}} - \xi_{j + 1}^{\prime {(n)}}} \right)}{M_{{K{({\xi_{j}^{(n)},\xi_{j + 1}^{(n)}})}},{T_{j}^{(n)}{(x)}}}}{K\left( {\xi_{j}^{(n)},\xi_{j + 1}^{(n)}} \right)}}}} & (19)\end{matrix}$

[0044] equation (4) is generalized as $\begin{matrix}\begin{matrix}{{f_{{lds},{RR}}^{(n)}(x)} = \quad {{g(x)} + {{T_{1}^{(n)}(x)}{g\left( \xi_{1}^{(n)} \right)}} +}} \\{\quad {{{T_{2}^{(n)}(x)}{g\left( \xi_{2}^{(n)} \right)}} +}} \\{\quad {{{T_{3}^{(n)}(x)}{g\left( \xi_{3}^{(n)} \right)}} + {\cdots \quad.}}}\end{matrix} & (20)\end{matrix}$

[0045] Equation (14) reduces to equation (4) in a manner similar to theway in which equation (11) reduces to equation (6) in the monochromaticcase. In comparison with equation (11), the g(ξ_(j) ^((n))) multipliersin the summands in equation (14) essentially represent descriptions ofthe flux of photons that are to be incident on a surface of an object ina scene, prior to their interaction with a respective surface of anobject in the scene onto which they are incident. The T_(J) ^((n)) inturn, represent the interactions of the photon flux with the respectivesurface, in particular how the respective surface reflects, absorbs,blocks, and so forth, respective ones of the photons of the variouswavelengths that comprise the flux that is incident on the respectivesurface.

[0046] Accordingly, the computer graphics system 10 can use equation(11) to approximate the evaluation the global illumination renderingequation (equation (7)) in connection with light source ray-to-eyetracing to render an image of a scene, or equation (14) to approximatethe evaluation of the global illumination rendering equation inconnection with eye-to-light source ray tracing.

[0047] In addition to accommodating a color spectrum, the improvedRussian Roulette methodology also accommodate other types ofcharacteristics for the surfaces defining the objects in the scene. Asan illustration of this, it will be assumed that a surface can bediffuse, glossy or specular, or any combination thereof. The totalphoton flux that is incident on a surface is represented by “f,” and, if“d,” “g” and “s” are functions that define the diffuseness, glossinessand specularity of the surface, respectively, the photon flux that isreflected by the surface will comprise three components, namely, adiffuse component “f·d,” a glossy component “f·g” and a specularcomponent “f·s.” Generally, the values of “f,” “d,” “g” and “s” can befunctions of the wavelength λ of the incident photon flux, and thecondition that the value of the norm of the kernel “K” not be greaterthan “one” requires that d_(λ)+g_(λ)+s_(λ)≦1 for all wavelengths λ. Inthat case, according to the Russian Roulette methodology, a photon'stracing path will be continued if, for the corresponding sample pointζ=ξ′_(j) ^((n)), $\begin{matrix}{ < \frac{\int_{0}^{\infty}{I_{\lambda}{f_{\lambda}\left( {d_{\lambda} + g_{\lambda} + s_{\lambda}} \right)}{\lambda}}}{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}{\lambda}}}} & (21)\end{matrix}$

[0048] The same sample point ζ can be used to obtain a decision as towhich of the interaction types is to be evaluated according to:$\begin{matrix}\left\{ \begin{matrix}{0 \leq  < \frac{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}d_{\lambda}{\lambda}}}{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}{\lambda}}}} & {{diffuse},{{flux}\quad {fd}{\int_{0}^{\infty}\frac{I_{\lambda}f_{\lambda}{\lambda}}{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}d_{\lambda}{\lambda}}}}}} \\{\frac{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}d_{\lambda}{\lambda}}}{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}{\lambda}}} \leq  < \frac{\int_{0}^{\infty}{I_{\lambda}{f_{\lambda}\left( {d_{\lambda} + g_{\lambda}} \right)}{\lambda}}}{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}{\lambda}}}} & {{glossy},{{flux}\quad {fg}{\int_{0}^{\infty}\frac{I_{\lambda}f_{\lambda}{\lambda}}{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}g_{\lambda}{\lambda}}}}}} \\{\frac{\int_{0}^{\infty}{I_{\lambda}{f_{\lambda}\left( {d_{\lambda} + g_{\lambda}} \right)}{\lambda}}}{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}{\lambda}}} \leq  < \frac{\int_{0}^{\infty}{I_{\lambda}{f_{\lambda}\left( {d_{\lambda} + g_{\lambda} + s_{\lambda}} \right)}{\lambda}}}{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}{\lambda}}}} & {{specular},{{flux}\quad {fs}{\int_{0}^{\infty}\frac{I_{\lambda}f_{\lambda}{\lambda}}{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}s_{\lambda}{\lambda}}}}}}\end{matrix} \right. & (22)\end{matrix}$

[0049] It will be appreciated that the particular order of diffuse,glossy and specular as appears in the respective lines of equation (16)may differ from the order in which they appear above. For example,instead of associating the diffuse interaction type and “d_(λ)” with thefirst line, the first line could be associated with the glossyinteraction type and “g_(λ)” or the specular interaction type and“s_(λ)” and similarly with the second and third lines. Extensions toother interaction types, in addition to or instead of the threedescribed here, will be apparent to those skilled in the art.

[0050] As noted above, color is typically represented by three colorchannels, such as separate red, blue, and green channels. If the colorsare given as f=(f_(r),f_(g),f_(b)), with “f_(r),” “f_(g)” and “f_(b)”representing the fluxes for the respective red, green and blue colors,and functions “d,” “g” and “s” are functions of the respective red,green and blue colors, d=(d_(r),d_(g),d_(b)), g=(g_(r),g_(g),g_(b)), ands=(s_(r),s_(g),s_(b)), the integrals in equation (16) are simply sums,such as, for example

∫₀ ^(∞) I _(λ) f _(λ) d _(λ) dλ=c _(r) f _(r) d _(r) +c _(g) f _(g) d_(g) +c _(b) f _(b) d _(b)  (23),

and

∫₀ ^(∞) I _(λ) f _(λ) dλ=c _(r) f _(r) +c _(g) f _(g) +c _(b) f_(b)  (24),

[0051] where “c_(r),” “c_(g),” and “c_(b)” are the color coefficients asdefined above.

[0052] The invention provides computer graphics system 10 that makes useof an improved Russian Roulette methodology that can provide formultichromatic color and various surface characteristics, such asglossy, diffuse and specular, of objects in a scene, in an imagerendered using the global illumination methodology.

[0053] It will be appreciated that numerous modifications may be made tothe arrangement described herein. For example, although the arrangementhas been described as making use of three color channels, namely, red,green and blue, it will be appreciated that other colors may be used inaddition to or instead of the three referenced here. In addition,although the arrangement has been described as making use of three typesof surface characteristics, namely, diffuse, glossy and specular, itwill be appreciated that other types of characteristics may be used inaddition to or instead of the three referenced here.

[0054] It will be appreciated that a system in accordance with theinvention can be constructed in whole or in part from special purposehardware or a general purpose computer system, or any combinationthereof, any portion of which may be controlled by a suitable program.Any program may in whole or in part comprise part of or be stored on thesystem in a conventional manner, or it may in whole or in part beprovided in to the system over a network or other mechanism fortransferring information in a conventional manner. In addition, it willbe appreciated that the system may be operated and/or otherwisecontrolled by means of information provided by an operator usingoperator input elements (not shown) which may be connected directly tothe system or which may transfer the information to the system over anetwork or other mechanism for transferring information in aconventional manner.

[0055] The foregoing description has been limited to a specificembodiment of this invention. It will be apparent, however, that variousvariations and modifications may be made to the invention, with theattainment of some or all of the advantages of the invention. It is theobject of the appended claims to cover these and such other variationsand modifications as come within the true spirit and scope of theinvention.

What is claimed as new and desired to be secured by Letters Patent ofthe United States is:
 1. A computer graphics system for generating apixel value for a pixel in an image, the pixel being representative of apoint {right arrow over (x)} in a scene, the computer graphics systemgenerating the pixel value to simulate global illumination for aplurality of colors, global illumination being represented by anevaluation of a function f(x), the function f(x) having the formf(x)=g(x)+∫₀ ¹K(x,y)f(y)dy, where f(x) (and similarly f(y)) is anunknown function, and g(x) and K(x,y) are known functions, with K(x,y)serving as a “kernel” of the integral, the kernel K(x,y) including afunction associated with at least two colors, the computer graphicssystem comprising: A. an estimator generator module configured togenerate a selected number “N” of estimators f^((n)) _(lds,RR)(x) as$\begin{matrix}{{{\overset{\_}{f}}_{{lds},{RR}}^{(n)}(x)} = \quad {{g(x)} + {{K\left( {x,\xi_{1}^{(n)}} \right)}S_{1}^{(n)}} +}} \\{\quad {{{K\left( {x,\xi_{2}^{(n)}} \right)}S_{2}^{(n)}} +}} \\{\quad {{{{K\left( {x,\xi_{3}^{(n)}} \right)}S_{3}^{(n)}} + \cdots}\quad,}}\end{matrix}$

 where S_(i) ^((n)) are defined recursively as${S_{1}^{(n)}:={g\left( \xi_{1}^{(n)} \right)}},{S_{j + 1}^{(n)}:={\frac{\Theta \left( {M_{{K{({\xi_{j + 1}^{n},\xi_{j}^{n}})}},S_{j}^{(n)}} - \xi_{j}^{\prime {(n)}}} \right)}{M_{{K{({\xi_{j + 1}^{n},\xi_{j}^{n}})}},S_{j}^{(n)}}}{K\left( {\xi_{j + 1}^{n},\xi_{j}^{n}} \right)}S_{j}^{(n)}}},{with}$$M_{A,B}:=\left\{ {\begin{matrix}{\frac{\int_{0}^{\infty}{A_{\lambda}B_{\lambda}{\lambda}}}{\int_{0}^{\infty}{B_{\lambda}{\lambda}}},} & {{{if}\quad {\int_{0}^{\infty}{B_{\lambda}{\lambda}}}} > 0} \\{0,} & {{{if}\quad {\int_{0}^{\infty}{B_{\lambda}{\lambda}}}} = 0}\end{matrix},} \right.$

 and where Θ(x) represents the Heaviside function, and ξ₁ ^((n)), ξ′₁^((n)), ξ₂ ^((n)), ξ′₂ ^((n)), . . . , ξ_(L) ^((n)), ξ′_(L) ^((n)),represent components of a predetermined multi-dimensional strictlydeterministic low-discrepancy sequence and where “n” denotes a sequenceindex, the estimator generator generating successive terms for eachestimator f^((n)) _(lds,RR)(x) until it generates a term having thevalue zero; B. a luminance value module configured to generate aluminance value for the point in the scene as the average of saidestimator values generated by said estimator generator module; and C. apixel value generator module configured to use the luminance valuegenerated for the point in the scene in generating the pixel value.
 2. Acomputer graphics system as defined in claim 1 in which the kernelK(x,y) further includes a function of at least one type of surfacecharacteristic of a surface in the scene, the estimator generator modulebeing configured to generate the estimator f^((n)) _(lds,RR)(x) inrelation to the at least one surface characteristic.
 3. A computergraphics system as defined in claim 2 in which the estimator generatormodule is configured to generate the estimator f^((n)) _(lds,RR)(x) inrelation to at least one photon, the estimator generator module beingconfigured to determine whether the at least one photon's path will becontinued in relation to${ < \frac{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}c_{\lambda}{\lambda}}}{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}{\lambda}}}},$

where I_(λ) is a weighting value as a function with which the human eyeperceives light having the respective wavelength λ, f_(λ) is the flux ofthe at least one photon per wavelength λ, c_(λ) is a function of therespective surface characteristic type as a function of the wavelengthλ, and ζ is a selected value.
 4. A computer graphics system as definedin claim 3 in which ζ is a function of ξ′_(j) ^((n)).
 5. A computergraphics system as defined in claim 3 in which the estimator generatormodule is configured to generate the estimator in relation to aplurality of surface characteristic types, each represented by afunction, c_(λ) being a sum of the respective functions.
 6. A computergraphics system as defined in claim 5 in which the estimator module isfurther configured to generate the estimator f^((n)) _(ldr,RR)(x) inrelation to an interaction type, the interaction type being related tothe respective surface characteristic types.
 7. A computer graphicssystem as defined in claim 6 in which the estimator module is configuredto generate the estimator f^((n)) _(lds,RR)(x) in relation to$\left\{ {\begin{matrix}{0 \leq  < \frac{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}d_{\lambda}{\lambda}}}{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}{\lambda}}}} & {{flux}\quad {fd}{\int_{0}^{\infty}\frac{I_{\lambda}f_{\lambda}{\lambda}}{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}d_{\lambda}{\lambda}}}}} \\{\frac{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}d_{\lambda}{\lambda}}}{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}{\lambda}}} \leq  < \frac{\int_{0}^{\infty}{I_{\lambda}{f_{\lambda}\left( {d_{\lambda} + g_{\lambda}} \right)}{\lambda}}}{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}{\lambda}}}} & {{flux}\quad {fg}{\int_{0}^{\infty}\frac{I_{\lambda}f_{\lambda}{\lambda}}{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}g_{\lambda}{\lambda}}}}} \\{\frac{\int_{0}^{\infty}{I_{\lambda}{f_{\lambda}\left( {d_{\lambda} + g_{\lambda}} \right)}{\lambda}}}{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}{\lambda}}} \leq  < \frac{\int_{0}^{\infty}{I_{\lambda}{f_{\lambda}\left( {d_{\lambda} + g_{\lambda} + s_{\lambda}} \right)}{\lambda}}}{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}{\lambda}}}} & {{flux}\quad {fs}{\int_{0}^{\infty}\frac{I_{\lambda}f_{\lambda}{\lambda}}{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}s_{\lambda}{\lambda}}}}} \\\cdots & \quad\end{matrix},} \right.$

for respective surface characteristic types associated with respectivefunctions d_(λ), g_(λ), s_(λ), . . . .
 8. A computer graphics system forgenerating a pixel value for a pixel in an image, the pixel beingrepresentative of a point {right arrow over (x)} in a scene, thecomputer graphics system generating the pixel value to simulate globalillumination for a plurality of colors, global illumination beingrepresented by an evaluation of a function f(x) over a sphere centeredat the point in the scene, the function f(x) having the formf(x)=g(x)+∫₀ ¹K(x,y)f(y)dy, where f(x) (and similarly f(y)) is anunknown function, and g(x) and K(x,y) are known functions, with K(x,y)serving as a “kernel” of the integral, the computer graphics systemcomprising: A. an estimator generator module configured to generate aselected number “N” of estimators f^((n)) _(lds,RR)(x) as$\begin{matrix}{{f_{{lds},{RR}}^{(n)}(x)} = \quad {{g(x)} + {{T_{1}^{(n)}(x)}{g\left( \xi_{1}^{(n)} \right)}} +}} \\{\quad {{{T_{2}^{(n)}(x)}{g\left( \xi_{2}^{(n)} \right)}} +}} \\{\quad {{{{T_{3}^{(n)}(x)}{g\left( \xi_{3}^{(n)} \right)}} + \cdots}\quad,}}\end{matrix}$

 where S_(i) ^((n)) are defined recursively as${T_{1}^{(n)}(x)}:={\frac{\Theta \left( {M_{{K{({x,\xi_{1}^{(n)}})}},1} - \xi_{1}^{\prime {(n)}}} \right)}{M_{{K{({x,\xi_{1}^{(n)}})}},1}}{K\left( {x,\xi_{1}^{(n)}} \right)}}$${{T_{j + 1}^{(n)}(x)}:={\frac{\Theta \left( {M_{{K{({\xi_{j}^{(n)},\xi_{j + 1}^{(n)}})}},{T_{j}^{(n)}{(x)}}} - \xi_{j + 1}^{\prime {(n)}}} \right)}{M_{{K{({\xi_{j}^{(n)},\xi_{j + 1}^{(n)}})}},{T_{j}^{(n)}{(x)}}}}{K\left( {\xi_{j}^{(n)},\xi_{j + 1}^{(n)}} \right)}}},{with}$$M_{A,B}:=\left\{ {\begin{matrix}{\frac{\int_{0}^{\infty}{A_{\lambda}B_{\lambda}{\lambda}}}{\int_{0}^{\infty}{B_{\lambda}{\lambda}}},} & {{{if}\quad {\int_{0}^{\infty}{B_{\lambda}{\lambda}}}} > 0} \\{0,} & {{{if}\quad {\int_{0}^{\infty}{B_{\lambda}{\lambda}}}} = 0}\end{matrix},} \right.$

 and where Θ(x) represents the Heaviside function, and ξ₁ ^((n)), ξ′₁^((n)), ξ₂ ^((n)), ξ′₂ ^((n)), . . . , ξ_(L) ^((n)), ξ′_(L) ^((n)),represent components of a predetermined multi-dimensional strictlydeterministic low-discrepancy sequence and where “n” denotes a sequenceindex, the estimator generator generating successive terms for eachestimator f^((n)) _(lds,RR)(x) until it generates a term having thevalue zero; B. a luminance value generator configured to generate aluminance value for the point in the scene as the average of saidestimator values generated by said estimator generator module; and C. apixel value generator module configured to use the luminance valuegenerated for the point in the scene in generating the pixel value.
 9. Acomputer graphics system as defined in claim 8 in which the kernelK(x,y) further includes a function of at least one type of surfacecharacteristic of a surface in the scene, the estimator generator moduleis configured to generate the estimator f^((n)) _(lds,RR)(x) in relationto the at least one surface characteristic.
 10. A computer graphicssystem as defined in claim 9 in which the estimator generator module isconfigured to generate the estimator f^((n)) _(lds,RR)(x) in relation toat least one photon, the estimator generator module being configured todetermine whether the at least one photon's path will be continued inrelation to${ < \frac{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}c_{\lambda}{\lambda}}}{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}{\lambda}}}},$

where I_(λ) is a weighting value as a function with which the human eyeperceives light having the respective wavelength λ, f_(λ) is the flux ofthe at least one photon per wavelength λ, c_(λ) is a function of therespective surface characteristic type as a function of the wavelengthλ, and ζ is a selected value.
 11. A computer graphics system as definedin claim 10 in which ζ is a function of ξ′_(j) ^((n)).
 12. A computergraphics system as defined in claim 10 in which the estimator generatormodule is configured to generate the estimator in relation to aplurality of surface characteristic types, each represented by afunction, c_(λ) being a sum of the respective functions.
 13. A computergraphics system as defined in claim 12 in which the estimator module isfurther configured to generate the estimator f^((n)) _(ldr,RR)(x) inrelation to an interaction type, the interaction type being related tothe respective surface characteristic types.
 14. A computer graphicssystem as defined in claim 13 in which the estimator module isconfigured to generate the estimator f^((n)) _(lds,RR)(x) in relation to$\left\{ {\begin{matrix}{0 \leq  < \frac{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}d_{\lambda}{\lambda}}}{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}{\lambda}}}} & {{flux}\quad {fd}\frac{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}{\lambda}}}{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}d_{\lambda}{\lambda}}}} \\{\frac{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}d_{\lambda}{\lambda}}}{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}{\lambda}}} \leq  < \frac{\int_{0}^{\infty}{I_{\lambda}{f_{\lambda}\left( {d_{\lambda} + g_{\lambda}} \right)}{\lambda}}}{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}{\lambda}}}} & {{flux}\quad {fg}\frac{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}{\lambda}}}{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}g_{\lambda}{\lambda}}}} \\{\frac{\int_{0}^{\infty}{I_{\lambda}{f_{\lambda}\left( {d_{\lambda} + g_{\lambda}} \right)}{\lambda}}}{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}{\lambda}}} \leq  < \frac{\int_{0}^{\infty}{I_{\lambda}{f_{\lambda}\left( {d_{\lambda} + g_{\lambda} + s_{\lambda}} \right)}{\lambda}}}{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}{\lambda}}}} & {{flux}\quad {fs}\frac{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}{\lambda}}}{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}s_{\lambda}{\lambda}}}} \\\cdots & \quad\end{matrix},} \right.$

for respective surface characteristic types associated with respectivefunctions d_(λ), g_(λ), s_(λ), . . . .
 15. A computer graphics methodfor generating a pixel value for a pixel in an image, the pixel beingrepresentative of a point {right arrow over (x)} in a scene, thecomputer graphics method generating the pixel value to simulate globalillumination for a plurality of colors, global illumination beingrepresented by an evaluation of a function f(x), the function f(x)having the form f(x)=g(x)+∫₀ ¹K(x,y)f(y)dy , where f(x) (and similarlyf(y)) is an unknown function, and g(x) and K(x,y) are known functions,with K(x,y) serving as a “kernel” of the integral, the kernel K(x,y)including a function associated with at least two colors, the computergraphics method comprising: A. an estimator generator step of generatinga selected number “N” of estimators f^((n)) _(lds,RR)(x) as$\begin{matrix}{{{\overset{\_}{f}}_{{lds},{RR}}^{(n)}(x)} = \quad {{g(x)} + {{K\left( {x,\xi_{1}^{(n)}} \right)}S_{1}^{(n)}} +}} \\{\quad {{{K\left( {x,\xi_{2}^{(n)}} \right)}S_{2}^{(n)}} +}} \\{\quad {{{{K\left( {x,\xi_{3}^{(n)}} \right)}S_{3}^{(n)}} + \cdots}\quad,}}\end{matrix}$

 where S₁ ^((n)) are defined recursively as${S_{1}^{(n)}:={g\left( \xi_{1}^{(n)} \right)}},{S_{j + 1}^{(n)}:={\frac{\Theta \left( {M_{{K{({\xi_{j + 1}^{n},\xi_{j}^{n}})}},S_{j}^{(n)}} - \xi_{j}^{\prime {(n)}}} \right)}{M_{{K{({\xi_{j + 1}^{n},\xi_{j}^{n}})}},S_{j}^{(n)}}}{K\left( {\xi_{j + 1}^{n},\xi_{j}^{n}} \right)}S_{j}^{(n)}}},{with}$$M_{A,B}:=\left\{ {\begin{matrix}{\frac{\int_{0}^{\infty}{A_{\lambda}B_{\lambda}{\lambda}}}{\int_{0}^{\infty}{B_{\lambda}{\lambda}}},} & {{{if}\quad {\int_{0}^{\infty}{B_{\lambda}{\lambda}}}} > 0} \\{0,} & {{{if}\quad {\int_{0}^{\infty}{B_{\lambda}{\lambda}}}} = 0}\end{matrix},} \right.$

 and where Θ(x) represents the Heaviside function, and ξ₁ ^((n)), ξ′₁^((n)), ξ₂ ^((n)), ξ′₂ ^((n)), . . . , ξ_(L) ^((n)), ξ′_(L) ^((n)),represent components of a predetermined multi-dimensional strictlydeterministic low-discrepancy sequence and where “n” denotes a sequenceindex, the estimator generator generating successive terms for eachestimator f^((n)) _(lds,RR)(x) until it generates a term having thevalue zero; B. a luminance value step of generating a luminance valuefor the point in the scene as the average of said estimator valuesgenerated during said estimator generator step; and C. a pixel valuegenerator step of using the luminance value generated for the point inthe scene in generating the pixel value.
 16. A computer graphics methodas defined in claim 15 in which the kernel K(x,y) further includes afunction of at least one type of surface characteristic of a surface inthe scene, the estimator generator step, including the step ofgenerating the estimator f^((n)) _(lds,RR)(x) in relation to the atleast one surface characteristic.
 17. A computer graphics method asdefined in claim 16 in which the estimator generator step includes thestep of generating the estimator f^((n)) _(lds,RR)(x) in relation to atleast one photon, the estimator generator step includes the step ofdetermining whether the at least one photon's path will be continued inrelation to${ < \frac{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}c_{\lambda}{\lambda}}}{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}{\lambda}}}},$

where I_(λ) is a weighting value as a function with which the human eyeperceives light having the respective wavelength λ, f_(λ) is the flux ofthe at least one photon per wavelength λ, c_(λ) is a function of therespective surface characteristic type as a function of the wavelengthλ, and ζ is a selected value.
 18. A computer graphics method as definedin claim 17 in which ζ is a function of ξ′_(j) ^((n)).
 19. A computergraphics method as defined in claim 17 in which the estimator generatorstep includes the step of generating the estimator in relation to aplurality of surface characteristic types, each represented by afunction, c_(λ) being a sum of the respective functions.
 20. A computergraphics method as defined in claim 19 in which the estimator stepincludes the step of generating the estimator f^((n)) _(ldr,RR)(x) inrelation to an interaction type, the interaction type being related tothe respective surface characteristic types.
 21. A computer graphicsmethod as defined in claim 20 in which the estimator step includes thestep of generating the estimator f^((n)) _(lds,RR)(x) in relation to$\left\{ {\begin{matrix}{0 \leq  < \frac{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}d_{\lambda}{\lambda}}}{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}{\lambda}}}} & {{flux}\quad {fd}\frac{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}{\lambda}}}{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}d_{\lambda}{\lambda}}}} \\{\frac{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}d_{\lambda}{\lambda}}}{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}{\lambda}}} \leq  < \frac{\int_{0}^{\infty}{I_{\lambda}{f_{\lambda}\left( {d_{\lambda} + g_{\lambda}} \right)}{\lambda}}}{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}{\lambda}}}} & {{flux}\quad {fg}\frac{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}{\lambda}}}{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}g_{\lambda}{\lambda}}}} \\{\frac{\int_{0}^{\infty}{I_{\lambda}{f_{\lambda}\left( {d_{\lambda} + g_{\lambda}} \right)}{\lambda}}}{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}{\lambda}}} \leq  < \frac{\int_{0}^{\infty}{I_{\lambda}{f_{\lambda}\left( {d_{\lambda} + g_{\lambda} + s_{\lambda}} \right)}{\lambda}}}{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}{\lambda}}}} & {{flux}\quad {fs}\frac{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}{\lambda}}}{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}s_{\lambda}{\lambda}}}} \\\cdots & \quad\end{matrix},} \right.$

for respective surface characteristic types associated with respectivefunctions d_(λ), g_(λ), s_(λ), . . . .
 22. A computer graphics methodfor generating a pixel value for a pixel in an image, the pixel beingrepresentative of a point {right arrow over (x)} in a scene, thecomputer graphics method generating the pixel value to simulate globalillumination for a plurality of colors, global illumination beingrepresented by an evaluation of a function f(x) over a sphere centeredat the point in the scene, the function f(x) having the formf(x)=g(x)+∫₀ ¹K(x,y)f(y)dy, where f(x) (and similarly f(y)) is anunknown function, and g(x) and K(x,y) are known functions, with K(x,y)serving as a “kernel” of the integral, the computer graphics methodcomprising: A. an estimator generator step of generating a selectednumber “N” of estimators f^((n)) _(lds,RR)(x) as $\begin{matrix}{{f_{{lds},{RR}}^{(n)}(x)} = \quad {{g(x)} + {{T_{1}^{(n)}(x)}{g\left( \xi_{1}^{(n)} \right)}} +}} \\{\quad {{{T_{2}^{(n)}(x)}{g\left( \xi_{2}^{(n)} \right)}} +}} \\{\quad {{{{T_{3}^{(n)}(x)}{g\left( \xi_{3}^{(n)} \right)}} + \cdots}\quad,}}\end{matrix}$

 where S₁ ^((n)) are defined recursively as${{T_{1}^{(n)}(x)}:={\frac{\Theta \left( {M_{{K{({x,\xi_{1}^{(n)}})}},1} - \xi_{1}^{\prime {(n)}}} \right)}{M_{{K{({x,\xi_{1}^{(n)}})}},1}}{K\left( {x,\xi_{1}^{(n)}} \right)}}},{{T_{j + 1}^{(n)}(x)}:={\frac{\Theta \left( {M_{{K{({\xi_{j}^{(n)},\xi_{j + 1}^{(n)}})}},{T_{j}^{(n)}{(x)}}} - \xi_{j + 1}^{\prime {(n)}}} \right)}{M_{{K{({\xi_{j}^{(n)},\xi_{j + 1}^{(n)}})}},{T_{j}^{(n)}{(x)}}}}{K\left( {\xi_{j}^{(n)},\xi_{j + 1}^{(n)}} \right)}}}$with $M_{A,B}:=\left\{ {\begin{matrix}{\frac{\int_{0}^{\infty}{A_{\lambda}B_{\lambda}{\lambda}}}{\int_{0}^{\infty}{B_{\lambda}{\lambda}}},} & {{{if}\quad {\int_{0}^{\infty}{B_{\lambda}{\lambda}}}} > 0} \\{0,} & {{{if}\quad {\int_{0}^{\infty}{B_{\lambda}{\lambda}}}} = 0}\end{matrix},} \right.$

 and where Θ(x) represents the Heaviside function, and ξ₁ ^((n)), ξ′₁^((n)), ξ₂ ^((n)), ξ′₂ ^((n)), . . . , ξ_(L) ^((n)), ξ′_(L) ^((n)),represent components of a predetermined multi-dimensional strictlydeterministic low-discrepancy sequence and where “n” denotes a sequenceindex, the estimator generator generating successive terms for eachestimator f^((n)) _(lds,RR)(x) until it generates a term having thevalue zero; B. a luminance value generator configured to generate aluminance value for the point in the scene as the average of saidestimator values generated during the estimator generator step; and C. apixel value generator step of using the luminance value generated forthe point in the scene in generating the pixel value.
 23. A computergraphics method as defined in claim 22 in which the kernel K(x,y)further includes a function of at least one type of surfacecharacteristic of a surface in the scene, the estimator generator stepincluding the step of generating the estimator f^((n)) _(lds,RR)(x) inrelation to the at least one surface characteristic.
 24. A computergraphics method as defined in claim 23 in which the estimator generatorstep includes the step of generating the estimator f^((n)) _(lds,RR)(x)in relation to at least one photon, the estimator generator stepincludes the step of determining whether the at least one photon's pathwill be continued in relation to${ < \frac{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}c_{\lambda}{\lambda}}}{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}{\lambda}}}},$

where I_(λ) is a weighting value as a function with which the human eyeperceives light having the respective wavelength λ, f_(λ) is the flux ofthe at least one photon per wavelength λ, c_(λ) is a function of therespective surface characteristic type as a function of the wavelengthλ, and ζ is a selected value.
 25. A computer graphics method as definedin claim 24 in which ζ is a function of ξ′_(j) ^((n)).
 26. A computergraphics method as defined in claim 24 in which the estimator generatorstep includes the step of generating the estimator in relation to aplurality of surface characteristic types, each represented by afunction, c_(λ) being a sum of the respective functions.
 27. A computergraphics method as defined in claim 26 in which the estimator stepincludes the step of generating the estimator f^((n)) _(ldr,RR)(x) inrelation to an interaction type, the interaction type being related tothe resepctive surface characteristic types.
 28. A computer graphicsmethod as defined in claim 27 in which the estimator step includes thestep of generating the estimator f^((n)) _(lds,RR)(x) in relation to$\left\{ {\begin{matrix}{0 \leq  < \frac{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}d_{\lambda}{\lambda}}}{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}{\lambda}}}} & {{flux}\quad {fd}\frac{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}{\lambda}}}{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}d_{\lambda}{\lambda}}}} \\{\frac{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}d_{\lambda}{\lambda}}}{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}{\lambda}}} \leq  < \frac{\int_{0}^{\infty}{I_{\lambda}{f_{\lambda}\left( {d_{\lambda} + g_{\lambda}} \right)}{\lambda}}}{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}{\lambda}}}} & {{flux}\quad {fg}\frac{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}{\lambda}}}{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}g_{\lambda}{\lambda}}}} \\{\frac{\int_{0}^{\infty}{I_{\lambda}{f_{\lambda}\left( {d_{\lambda} + g_{\lambda}} \right)}{\lambda}}}{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}{\lambda}}} \leq  < \frac{\int_{0}^{\infty}{I_{\lambda}{f_{\lambda}\left( {d_{\lambda} + g_{\lambda} + s_{\lambda}} \right)}{\lambda}}}{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}{\lambda}}}} & {{flux}\quad {fs}\frac{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}{\lambda}}}{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}s_{\lambda}{\lambda}}}} \\\cdots & \quad\end{matrix},} \right.$

for respective surface characteristic types associated with respectivefunctions d_(λ), g_(λ), s_(λ), . . . .
 29. A computer program productfor use in connection with a computer to generate a pixel value for apixel in an image, the pixel being representative of a point {rightarrow over (x)} in a scene, the computer graphics system generating thepixel value to simulate global illumination for a plurality of colors,global illumination being represented by an evaluation of a functionf(x), the function f(x) having the form f(x)=g(x)+∫₀ ¹K(x,y)f(y)dy,where f(x) (and similarly f(y)) is an unknown function, and g(x) andK(x,y) are known functions, with K(x,y) serving as a “kernel” of theintegral, the kernel K(x,y) including a function associated with atleast two colors, the computer program product comprising acomputer-readable medium having encoded thereon: A. an estimatorgenerator module configure to enable the computer to generate a selectednumber “N” of estimators f^((n)) _(lds,RR)(x) as $\begin{matrix}{{{\overset{\_}{f}}_{{lds},{RR}}^{(n)}(x)} = \quad {{g(x)} + {{K\left( {x,\xi_{1}^{(n)}} \right)}S_{1}^{(n)}} +}} \\{\quad {{{K\left( {x,\xi_{2}^{(n)}} \right)}S_{2}^{(n)}} +}} \\{\quad {{{{K\left( {x,\xi_{3}^{(n)}} \right)}S_{3}^{(n)}} + \cdots}\quad,}}\end{matrix}$

 where S₁ ^((n)) are defined recursively as${S_{1}^{(n)}:{g\left( \xi_{1}^{(n)} \right)}},{S_{j + 1}^{(n)}:={\frac{\Theta \left( {M_{{K{({\xi_{j + 1}^{n},\xi_{j}^{n}})}},S_{j}^{(n)}} - \xi_{j}^{\prime {(n)}}} \right)}{M_{{K{({\xi_{j + 1}^{n},\xi_{j}^{n}})}},S_{j}^{(n)}}}{K\left( {\xi_{j + 1}^{n},\xi_{j}^{n}} \right)}S_{j}^{(n)}}},{with}$

$M_{A,B}:=\left\{ {\begin{matrix}{\frac{\int_{0}^{\infty}{A_{\lambda}B_{\lambda}{\lambda}}}{\int_{0}^{\infty}{B_{\lambda}{\lambda}}},} & {{{if}\quad {\int_{0}^{\infty}{B_{\lambda}{\lambda}}}} > 0} \\{0,} & {{{if}\quad {\int_{0}^{\infty}{B_{\lambda}{\lambda}}}} = 0}\end{matrix},} \right.$

 and where Θ(x) represents the Heaviside function, and ξ₁ ^((n)), ξ′₁^((n)), ξ₂ ^((n)), ξ′₂ ^((n)), . . . , ξ_(L) ^((n)), ξ′_(L) ^((n)),represent components of a predetermined multi-dimensional strictlydeterministic low-discrepancy sequence and where “n” denotes a sequenceindex, the estimator generator generating successive terms for eachestimator f^((n)) _(lds,RR)(x) until it generates a term having thevalue zero; B. a luminance value module configured to generate aluminance value for the point in the scene as the average of saidestimator values generated by said estimator generator module; and C. apixel value generator module configured to use the luminance valuegenerated for the point in the scene in generating the pixel value. 30.A computer program product as defined in claim 29 in which the kernelK(x,y) further includes a function of at least one type of surfacecharacteristic of a surface in the scene, the estimator generator moduleis configured to generate the estimator f^((n)) _(lds,RR)(x) in relationto the at least one surface characteristic.
 31. A computer programproduct as defined in claim 30 in which the estimator generator moduleis configured to generate the estimator f^((n)) _(lds,RR)(X) in relationto at least one photon, the estimator generator module being configuredto determine whether the at least one photon's path will be continued inrelation to${ < \frac{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}c_{\lambda}{\lambda}}}{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}{\lambda}}}},$

where I_(λ) is a weighting value as a function with which the human eyeperceives light having the respective wavelength λ, f_(λ) is the flux ofthe at least one photon per wavelength λ, c_(λ) is a function of therespective surface characteristic type as a function of the wavelengthλ, and ζ is a selected value.
 32. A computer program product as definedin claim 31 in which ζ is a function of ξ′_(j) ^((n)).
 33. A computerprogram product as defined in claim 31 in which the estimator generatormodule is configured to generate the estimator in relation to aplurality of surface characteristic types, each represented by afunction, c_(λ) being a sum of the respective functions.
 34. A computerprogram product as defined in claim 33 in which the estimator stepincludes the step of generating the estimator f^((n)) _(ldr,RR)(x) inrelation to an interaction type, the interaction type being related tothe respective surface characteristic types.
 35. A computer programproduct as defined in claim 34 in which the estimator module isconfigured to generate the estimator f^((n)) _(lds,RR)(x) in relation to$\left\{ {\begin{matrix}{0 \leq  < \frac{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}d_{\lambda}{\lambda}}}{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}{\lambda}}}} & {{flux}\quad {fd}\frac{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}{\lambda}}}{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}d_{\lambda}{\lambda}}}} \\{\frac{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}d_{\lambda}{\lambda}}}{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}{\lambda}}} \leq  < \frac{\int_{0}^{\infty}{I_{\lambda}{f_{\lambda}\left( {d_{\lambda} + g_{\lambda}} \right)}{\lambda}}}{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}{\lambda}}}} & {{flux}\quad {fg}\frac{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}{\lambda}}}{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}g_{\lambda}{\lambda}}}} \\{\frac{\int_{0}^{\infty}{I_{\lambda}{f_{\lambda}\left( {d_{\lambda} + g_{\lambda}} \right)}{\lambda}}}{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}{\lambda}}} \leq  < \frac{\int_{0}^{\infty}{I_{\lambda}{f_{\lambda}\left( {d_{\lambda} + g_{\lambda} + s_{\lambda}} \right)}{\lambda}}}{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}{\lambda}}}} & {{flux}\quad {fs}\frac{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}{\lambda}}}{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}s_{\lambda}{\lambda}}}} \\\cdots & \quad\end{matrix},} \right.$

for respective surface characteristic types associated with respectivefunctions d_(λ), g_(λ), s_(λ), . . . .
 36. A computer program productfor generating a pixel value for a pixel in an image, the pixel beingrepresentative of a point {right arrow over (x)} in a scene, thecomputer graphics system generating the pixel value to simulate globalillumination for a plurality of colors, global illumination beingrepresented by an evaluation of a function f(x) over a sphere centeredat the point in the scene, the function f(x) having the formf(x)=g(x)+∫₀ ¹K(x,y)f(y)dy, where f(x) (and similarly f(y)) is anunknown function, and g(x) and K(x,y) are known functions, with K(x,y)serving as a “kernel” of the integral, the computer program productcomprising a computer-readable medium having encoded thereon: A. anestimator generator module configured to generate a selected number “N”of estimators f^((n)) _(lds,RR)(X) as $\begin{matrix}{{f_{{lds},{RR}}^{(n)}(x)} = \quad {{g(x)} + {{T_{1}^{(n)}(x)}{g\left( \xi_{1}^{(n)} \right)}} +}} \\{\quad {{{T_{2}^{(n)}(x)}{g\left( \xi_{2}^{(n)} \right)}} +}} \\{\quad {{{{T_{3}^{(n)}(x)}{g\left( \xi_{3}^{(n)} \right)}} + \cdots}\quad,}}\end{matrix}$

 where S₁ ^((n)) are defined recursively as${{T_{1}^{(n)}(x)}:={\frac{\Theta \left( {M_{{K{({x,\xi_{1}^{(n)}})}},1} - \xi_{1}^{\prime {(n)}}} \right)}{M_{{K{({x,\xi_{1}^{(n)}})}},1}}{K\left( {x,\xi_{1}^{(n)}} \right)}}},{{T_{j + 1}^{(n)}(x)}:={\frac{\Theta \left( {M_{{K{({\xi_{j}^{(n)},\xi_{j + 1}^{(n)}})}},{T_{j}^{(n)}{(x)}}} - \xi_{j + 1}^{\prime {(n)}}} \right)}{M_{{K{({\xi_{j}^{(n)},\xi_{j + 1}^{(n)}})}},{T_{j}^{(n)}{(x)}}}}K\left( {x,\xi_{j}^{(n)},\xi_{j + 1}^{(n)}} \right)}}$with $M_{A,B}:=\left\{ {\begin{matrix}{\frac{\int_{0}^{\infty}{A_{\lambda}B_{\lambda}{\lambda}}}{\int_{0}^{\infty}{B_{\lambda}{\lambda}}},} & {{{if}\quad {\int_{0}^{\infty}{B_{\lambda}{\lambda}}}} > 0} \\{0,} & {{{if}\quad {\int_{0}^{\infty}{B_{\lambda}{\lambda}}}} = 0}\end{matrix},} \right.$

 and where Θ(x) represents the Heaviside function, and ξ₁ ^((n)), ξ′₁^((n)), ξ₂ ^((n)), ξ′₂ ^((n)), . . . , ξ_(L) ^((n)), ξ′_(L) ^((n)),represent components of a predetermined multi-dimensional strictlydeterministic low-discrepancy sequence and where “n” denotes a sequenceindex, the estimator generator generating successive terms for eachestimator f^((n)) _(lds,RR)(x) until it generates a term having thevalue zero; B. a luminance value generator configured to generate aluminance value for the point in the scene as the average of saidestimator values generated by said estimator generator module; and C. apixel value generator module configured to use the luminance valuegenerated for the point in the scene in generating the pixel value. 37.A computer program product as defined in claim 36 in which the kernelK(x,y) further includes a function of at least one type of surfacecharacteristic of a surface in the scene, the estimator generator moduleis configured to generate the estimator f^((n)) _(lds,RR)(x) in relationto the at least one surface characteristic.
 38. A computer programproduct as defined in claim 37 in which the estimator generator moduleis configured to generate the estimator f^((n)) _(lds,RR)(x) in relationto at least one photon, the estimator generator module being configuredto determine whether the at least one photon's path will be continued inrelation to${ < \frac{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}c_{\lambda}{\lambda}}}{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}{\lambda}}}},$

where I_(λ) is a weighting value as a function with which the human eyeperceives light having the respective wavelength λ, f_(λ) is the flux ofthe at least one photon per wavelength λ, c_(λ) is a function of therespective surface characteristic type as a function of the wavelengthλ, and ζ is a selected value.
 39. A computer program product as definedin claim 38 in which ζ is a function of ξ′_(j) ^((n)).
 40. A computerprogram product as defined in claim 38 in which the estimator generatormodule is configured to generate the estimator in relation to aplurality of surface characteristic types, each represented by afunction, c_(λ) being a sum of the respective functions.
 41. A computerprogram product as defined in claim 40 in which the estimator stepincludes the step of generating the estimator f^((n)) _(ldr,RR)(x) inrelation to an interaction type, the interaction type being related tothe respective surface characteristic types.
 42. A computer programproduct as defined in claim 41 in which the estimator module isconfigured to generate the estimator f^((n)) _(lds,RR)(x) in relation to$\left\{ {\begin{matrix}{0 \leq  < \frac{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}d_{\lambda}{\lambda}}}{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}{\lambda}}}} & {{flux}\quad {fd}\frac{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}{\lambda}}}{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}d_{\lambda}{\lambda}}}} \\{\frac{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}d_{\lambda}{\lambda}}}{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}{\lambda}}} \leq  < \frac{\int_{0}^{\infty}{I_{\lambda}{f_{\lambda}\left( {d_{\lambda} + g_{\lambda}} \right)}{\lambda}}}{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}{\lambda}}}} & {{flux}\quad {fg}\frac{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}{\lambda}}}{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}g_{\lambda}{\lambda}}}} \\{\frac{\int_{0}^{\infty}{I_{\lambda}{f_{\lambda}\left( {d_{\lambda} + g_{\lambda}} \right)}{\lambda}}}{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}{\lambda}}} \leq  < \frac{\int_{0}^{\infty}{I_{\lambda}{f_{\lambda}\left( {d_{\lambda} + g_{\lambda} + s_{\lambda}} \right)}{\lambda}}}{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}{\lambda}}}} & {{flux}\quad {fs}\frac{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}{\lambda}}}{\int_{0}^{\infty}{I_{\lambda}f_{\lambda}s_{\lambda}{\lambda}}}} \\\cdots & \quad\end{matrix},} \right.$

for respective surface characteristic types associated with respectivefunctions d_(λ), g_(λ), s_(λ), . . . .